Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. $$ f(x)=\frac{3+x}{x^{3}-27} $$

Step-by-Step Solution

Answer

Domain: all real numbers except \( x = 3 \). Vertical asymptote at \( x = 3 \). Horizontal asymptote at \( y = 0 \).

Step 1: Find the Domain

The domain of the function is the set of all real numbers, except for those that make the denominator equal to zero. Calculate the values of \( x \) that make \( x^{3}-27 = 0 \). Solving for \( x \), we get: \( x^{3} = 27 \). This means \( x = 3 \). So, the domain is all real numbers except \( x = 3 \).

Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur where the function's denominator is zero and the numerator is non-zero at that point. From Step 1, we know \( x = 3 \) makes the denominator zero. The numerator, \( 3 + x \), is not zero at \( x = 3 \) since \( 3+3 eq 0 \). Therefore, there is a vertical asymptote at \( x = 3 \).

Step 3: Find Horizontal Asymptotes

Horizontal asymptotes depend on the degrees of the polynomial in the numerator and denominator. The degree of the numerator (\( 1 \)) is less than the degree of the denominator (\( 3 \)). This indicates the horizontal asymptote is at \( y = 0 \).

Key Concepts

Vertical Asymptotes

Vertical asymptotes signify where a function approaches infinity and are found where the denominator of a fraction equals zero. To find these critical points, solve for values that invalidate the denominator. For our example, the function is \( f(x) = \frac{3+x}{x^3 - 27} \). We are interested in when the function's denominator, \( x^3 - 27 \), equals zero. Setting it equal to zero gives us the equation: \[ x^3 - 27 = 0 \]Upon solving, we find \( x = 3 \). To confirm a vertical asymptote exists at \( x = 3 \), check that the numerator is non-zero at this point. Since \( 3 + 3 eq 0 \), a vertical asymptote definitely exists at \( x = 3 \). Remember, functions may have more than one vertical asymptote depending on their complexity.

Horizontal Asymptotes

Horizontal asymptotes illustrate the behavior of a function as \( x \) approaches infinity or negative infinity. These are determined by comparing the degrees of the terms in the numerator and denominator.In our function \( f(x) = \frac{3+x}{x^3 - 27} \), the degree of the numerator is 1, while the degree of the denominator is 3. Since the degree of the denominator exceeds that of the numerator, we conclude that the horizontal asymptote is \( y = 0 \).This is a common rule: if the denominator's degree is greater than the numerator's degree, the horizontal asymptote will always be \( y = 0 \). Understanding this rule can help you quickly determine the horizontal asymptote for similar functions without extensive calculations.

Domain of a Function

The domain of a function represents all possible input values (\( x \)) for which the function is defined. For rational functions like \( f(x) = \frac{3+x}{x^3 - 27} \), this means identifying values of \( x \) that do not cause the denominator to equal zero.In our case, we solve \( x^3 - 27 = 0 \) to find \( x = 3 \). This value makes the denominator undefined, so it must be excluded from the domain.Thus, the domain consists of all real numbers except \( x = 3 \).In general, for rational functions, always identify points where the denominator equals zero and exclude these points from the domain to avoid undefined expressions.